Optimal. Leaf size=245 \[ -\frac {3 (a+b x)^4}{128 b}+\frac {51 (a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {\left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^3}{8 b}-\frac {3 \left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)}{32 b}-\frac {45 \sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{64 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {27 \sin ^{-1}(a+b x)^2}{128 b} \]
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Rubi [A] time = 0.32, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4807, 4649, 4647, 4641, 4627, 4707, 30, 4677, 14} \[ -\frac {3 (a+b x)^4}{128 b}+\frac {51 (a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {\left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^3}{8 b}-\frac {3 \left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)}{32 b}-\frac {45 \sqrt {1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{64 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {27 \sin ^{-1}(a+b x)^2}{128 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 4627
Rule 4641
Rule 4647
Rule 4649
Rule 4677
Rule 4707
Rule 4807
Rubi steps
\begin {align*} \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}-\frac {3 \operatorname {Subst}\left (\int x \left (1-x^2\right ) \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{4 b}+\frac {3 \operatorname {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{4 b}\\ &=\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}-\frac {3 \operatorname {Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{8 b}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {9 \operatorname {Subst}\left (\int x \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{8 b}\\ &=-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}+\frac {3 \operatorname {Subst}\left (\int x \left (1-x^2\right ) \, dx,x,a+b x\right )}{32 b}-\frac {9 \operatorname {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{32 b}+\frac {9 \operatorname {Subst}\left (\int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=-\frac {45 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{64 b}-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}+\frac {3 \operatorname {Subst}\left (\int \left (x-x^3\right ) \, dx,x,a+b x\right )}{32 b}+\frac {9 \operatorname {Subst}(\int x \, dx,x,a+b x)}{64 b}-\frac {9 \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{64 b}+\frac {9 \operatorname {Subst}(\int x \, dx,x,a+b x)}{16 b}+\frac {9 \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{16 b}\\ &=\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}-\frac {45 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{64 b}-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}+\frac {27 \sin ^{-1}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 272, normalized size = 1.11 \[ \frac {3 \left (17-6 a^2\right ) b^2 x^2+6 a \left (17-2 a^2\right ) b x-16 \sqrt {-a^2-2 a b x-b^2 x^2+1} \left (2 a^3+6 a^2 b x+6 a b^2 x^2-5 a+2 b^3 x^3-5 b x\right ) \sin ^{-1}(a+b x)^3+6 \sqrt {-a^2-2 a b x-b^2 x^2+1} \left (2 a^3+6 a^2 b x+6 a b^2 x^2-17 a+2 b^3 x^3-17 b x\right ) \sin ^{-1}(a+b x)+3 \left (8 a^4+32 a^3 b x+8 a^2 \left (6 b^2 x^2-5\right )+16 a b x \left (2 b^2 x^2-5\right )+8 b^4 x^4-40 b^2 x^2+17\right ) \sin ^{-1}(a+b x)^2-12 a b^3 x^3+12 \sin ^{-1}(a+b x)^4-3 b^4 x^4}{128 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 243, normalized size = 0.99 \[ -\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} - 17\right )} b^{2} x^{2} - 12 \, \arcsin \left (b x + a\right )^{4} + 6 \, {\left (2 \, a^{3} - 17 \, a\right )} b x - 3 \, {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} - 5 \, a\right )} b x - 40 \, a^{2} + 17\right )} \arcsin \left (b x + a\right )^{2} + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (8 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 17\right )} b x - 17 \, a\right )} \arcsin \left (b x + a\right )\right )}}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.29, size = 296, normalized size = 1.21 \[ \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{4 \, b} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{8 \, b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )^{2}}{16 \, b} + \frac {3 \, \arcsin \left (b x + a\right )^{4}}{32 \, b} - \frac {3 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{32 \, b} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{16 \, b} - \frac {45 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{64 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{128 \, b} - \frac {45 \, \arcsin \left (b x + a\right )^{2}}{128 \, b} + \frac {45 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{128 \, b} + \frac {189}{1024 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 628, normalized size = 2.56 \[ \frac {-12+80 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x b -240 \arcsin \left (b x +a \right )^{2} x a b -102 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x b -32 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a^{3}+12 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a^{3}-12 x^{3} a \,b^{3}-18 x^{2} a^{2} b^{2}-12 x \,a^{3} b -96 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} x \,a^{2} b +36 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) x^{2} a \,b^{2}+36 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) x \,a^{2} b -120 \arcsin \left (b x +a \right )^{2} a^{2}+51 \arcsin \left (b x +a \right )^{2}+51 b^{2} x^{2}+102 a b x -96 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} x^{2} a \,b^{2}+51 a^{2}-120 \arcsin \left (b x +a \right )^{2} x^{2} b^{2}+12 \arcsin \left (b x +a \right )^{4}+24 \arcsin \left (b x +a \right )^{2} x^{4} b^{4}+96 \arcsin \left (b x +a \right )^{2} x^{3} a \,b^{3}+144 \arcsin \left (b x +a \right )^{2} x^{2} a^{2} b^{2}+96 \arcsin \left (b x +a \right )^{2} x \,a^{3} b -32 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} x^{3} b^{3}+12 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) x^{3} b^{3}+80 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -102 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -3 x^{4} b^{4}+24 \arcsin \left (b x +a \right )^{2} a^{4}-3 a^{4}}{128 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {asin}\left (a+b\,x\right )}^3\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.20, size = 694, normalized size = 2.83 \[ \begin {cases} \frac {3 a^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} + \frac {3 a^{3} x \operatorname {asin}^{2}{\left (a + b x \right )}}{4} - \frac {3 a^{3} x}{32} - \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4 b} + \frac {3 a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32 b} + \frac {9 a^{2} b x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{8} - \frac {9 a^{2} b x^{2}}{64} - \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {9 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 a^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} + \frac {3 a b^{2} x^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} - \frac {3 a b^{2} x^{3}}{32} - \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {9 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 a x \operatorname {asin}^{2}{\left (a + b x \right )}}{8} + \frac {51 a x}{64} + \frac {5 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{8 b} - \frac {51 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{64 b} + \frac {3 b^{3} x^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{16} - \frac {3 b^{3} x^{4}}{128} - \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {3 b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 b x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{16} + \frac {51 b x^{2}}{128} + \frac {5 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{8} - \frac {51 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{64} + \frac {3 \operatorname {asin}^{4}{\left (a + b x \right )}}{32 b} + \frac {51 \operatorname {asin}^{2}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac {3}{2}} \operatorname {asin}^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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